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<a href="rotations_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno">    1</span><span class="comment">/*!</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno">    2</span><span class="comment"> * </span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno">    3</span><span class="comment"> *</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno">    4</span><span class="comment"> * \brief       Some operations for creating rotation matrices</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno">    5</span><span class="comment"> * </span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno">    6</span><span class="comment"> * </span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno">    7</span><span class="comment"> * </span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno">    8</span><span class="comment"> *</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno">    9</span><span class="comment"> * \author      O. Krause</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno">   10</span><span class="comment"> * \date        2011</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="comment"> *</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span><span class="comment"> *</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="comment"> * \par Copyright 1995-2017 Shark Development Team</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="comment"> * </span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span><span class="comment"> * &lt;BR&gt;&lt;HR&gt;</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno">   16</span><span class="comment"> * This file is part of Shark.</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno">   17</span><span class="comment"> * &lt;https://shark-ml.github.io/Shark/&gt;</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno">   18</span><span class="comment"> * </span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno">   19</span><span class="comment"> * Shark is free software: you can redistribute it and/or modify</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno">   20</span><span class="comment"> * it under the terms of the GNU Lesser General Public License as published </span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno">   21</span><span class="comment"> * by the Free Software Foundation, either version 3 of the License, or</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno">   22</span><span class="comment"> * (at your option) any later version.</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno">   23</span><span class="comment"> * </span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno">   24</span><span class="comment"> * Shark is distributed in the hope that it will be useful,</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno">   25</span><span class="comment"> * but WITHOUT ANY WARRANTY; without even the implied warranty of</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno">   26</span><span class="comment"> * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno">   27</span><span class="comment"> * GNU Lesser General Public License for more details.</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span><span class="comment"> * </span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span><span class="comment"> * You should have received a copy of the GNU Lesser General Public License</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span><span class="comment"> * along with Shark.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span><span class="comment"> *</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span><span class="comment"> */</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span> </div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno">   34</span><span class="preprocessor">#ifndef SHARK_LINALG_ROTATIONS_H</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno">   35</span><span class="preprocessor">#define SHARK_LINALG_ROTATIONS_H</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno">   36</span> </div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno">   37</span><span class="preprocessor">#include &lt;<a class="code" href="_base_8h.html">shark/LinAlg/Base.h</a>&gt;</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno">   38</span><span class="preprocessor">#include &lt;<a class="code" href="_random_8h.html">shark/Core/Random.h</a>&gt;</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno">   39</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespaceshark.html" title="AbstractMultiObjectiveOptimizer.">shark</a>{ <span class="keyword">namespace </span>blas{<span class="comment"></span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno">   40</span><span class="comment">/**</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno">   41</span><span class="comment"> * \ingroup shark_globals</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno">   42</span><span class="comment"> * </span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno">   43</span><span class="comment"> * @{</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno">   44</span><span class="comment"> */</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno">   45</span><span class="comment"></span> </div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno">   46</span><span class="comment">//! \brief Generates a Householder reflection from a vector to use with applyHouseholderLeft/Right</span></div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span><span class="comment">//!</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span><span class="comment">//! Given a Vector x=(x0,x1,...,xn), finds a reflection with the property</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span><span class="comment">//! (c, 0,0,...0) = (I-beta v v^t)x</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno">   50</span><span class="comment">//! and v = (x0-c,x1,x2,...,xn)</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno">   51</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> X, <span class="keyword">class</span> R&gt;</div>
<div class="foldopen" id="foldopen00052" data-start="{" data-end="}">
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"><a class="line" href="group__shark__globals.html#ga44d754a05d75cb6c40c0c292f3e89ab8">   52</a></span><span class="keyword">typename</span> X::value_type <a class="code hl_function" href="group__shark__globals.html#ga44d754a05d75cb6c40c0c292f3e89ab8" title="Generates a Householder reflection from a vector to use with applyHouseholderLeft/Right.">createHouseholderReflection</a>(</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno">   53</span>    vector_expression&lt;X, cpu_tag&gt; <span class="keyword">const</span>&amp; x, </div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno">   54</span>    vector_expression&lt;R, cpu_tag&gt;&amp; reflection</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno">   55</span>){</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno">   56</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(x().size() != 0);</div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno">   57</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(x().size() == reflection().size());</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno">   58</span>    </div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno">   59</span>    <span class="comment">//special case for last iteration of QR etc</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span>    <span class="comment">//by convention diagonal elements are &gt; 0</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span>    <span class="keywordflow">if</span>(x().size() == 1){</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span>        reflection()(0) = 1;</div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno">   63</span>        <span class="keywordflow">return</span> 2;</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno">   64</span>    }</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno">   65</span>        </div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno">   66</span>    </div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno">   67</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> X::value_type Value;</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno">   68</span>    </div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno">   69</span>    <span class="keywordtype">double</span> norm = norm_2(x);</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno">   70</span>    <span class="keywordflow">if</span> (x()(0) &gt;= 0.0)</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno">   71</span>        norm *= -1.0;</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno">   72</span>    </div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno">   73</span>    noalias(reflection()) = x;</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno">   74</span>    reflection()(0) -= norm;</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span>    reflection() /= (x()(0) - norm);</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span>    <span class="comment">//if pivot is close to 0, this is one-&gt;numericaly stable</span></div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span>    <span class="comment">//compared to the usual formula</span></div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span>    Value beta = (norm - x()(0)) / norm;</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno">   79</span>    <span class="keywordflow">return</span> beta;</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno">   80</span>}</div>
</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno">   81</span><span class="comment">//\brief rotates a matrix using a householder reflection </span></div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno">   82</span><span class="comment">//</span></div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno">   83</span><span class="comment">//calculates A*(1-beta*xx^T)</span></div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno">   84</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> Mat, <span class="keyword">class</span> R, <span class="keyword">class</span> T, <span class="keyword">class</span> Device&gt;</div>
<div class="foldopen" id="foldopen00085" data-start="{" data-end="}">
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"><a class="line" href="group__shark__globals.html#gabead541306577f124c02ac43c10100a7">   85</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="group__shark__globals.html#gabead541306577f124c02ac43c10100a7">applyHouseholderOnTheRight</a>(</div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno">   86</span>    matrix_expression&lt;Mat, Device&gt; &amp; matrix,</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno">   87</span>    vector_expression&lt;R, Device&gt; <span class="keyword">const</span>&amp; reflection, </div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno">   88</span>    T beta</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno">   89</span>){</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno">   90</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(matrix().size2() == reflection().size());</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno">   91</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(reflection().size() != 0 );</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno">   92</span>    </div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno">   93</span>    <span class="comment">//special case for last iteration of QR etc</span></div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno">   94</span>    <span class="keywordflow">if</span>(reflection().size() == 1){</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno">   95</span>        matrix() *= 1-beta;</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno">   96</span>        <span class="keywordflow">return</span>;</div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno">   97</span>    }</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno">   98</span>    </div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno">   99</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(matrix().size2() == reflection().size());</div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno">  100</span>    <span class="comment">//Ax</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno">  101</span>    blas::vector&lt;T&gt; temp = prod(matrix,reflection);</div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno">  102</span>    </div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno">  103</span>    <span class="comment">//A -=beta*(Ax)x^T</span></div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno">  104</span>    noalias(matrix()) -= beta * outer_prod(temp,reflection);</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno">  105</span>}</div>
</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno">  106</span> </div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno">  107</span><span class="comment"></span> </div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno">  108</span><span class="comment">/// \brief rotates a matrix using a householder reflection </span></div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno">  109</span><span class="comment">///</span></div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno">  110</span><span class="comment">/// calculates (1-beta*xx^T)*A</span></div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno">  111</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> Mat, <span class="keyword">class</span> R, <span class="keyword">class</span> T, <span class="keyword">class</span> Device&gt;</div>
<div class="foldopen" id="foldopen00112" data-start="{" data-end="}">
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"><a class="line" href="group__shark__globals.html#gaf437ca980035fd6fb152bcfa0cd34d82">  112</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="group__shark__globals.html#gaf437ca980035fd6fb152bcfa0cd34d82" title="rotates a matrix using a householder reflection">applyHouseholderOnTheLeft</a>(</div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno">  113</span>    matrix_expression&lt;Mat, Device&gt; &amp; matrix,</div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno">  114</span>    vector_expression&lt;R, Device&gt; <span class="keyword">const</span>&amp; reflection, </div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno">  115</span>    T <span class="keyword">const</span>&amp; beta</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno">  116</span>){</div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno">  117</span> </div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno">  118</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(matrix().size1() == reflection().size());</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno">  119</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(reflection().size() != 0 );</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno">  120</span>    </div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno">  121</span>    <span class="comment">//special case for last iteration of QR etc</span></div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno">  122</span>    <span class="keywordflow">if</span>(reflection().size() == 1){</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno">  123</span>        matrix()*=1-beta;</div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno">  124</span>        <span class="keywordflow">return</span>;</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno">  125</span>    }</div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno">  126</span>    <span class="comment">//x^T A</span></div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno">  127</span>    blas::vector&lt;T&gt; temp = prod(trans(matrix),reflection);</div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno">  128</span>    </div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno">  129</span>    <span class="comment">//A -=beta*x(x^T A)</span></div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno">  130</span>    noalias(matrix()) -= beta * outer_prod(reflection,temp);</div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno">  131</span>}</div>
</div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno">  132</span><span class="comment"></span> </div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno">  133</span><span class="comment">/// \brief rotates a matrix using a householder reflection </span></div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno">  134</span><span class="comment">///</span></div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno">  135</span><span class="comment">/// calculates (1-beta*xx^T)*A</span></div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno">  136</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> Mat, <span class="keyword">class</span> R, <span class="keyword">class</span> T, <span class="keyword">class</span> Device&gt;</div>
<div class="foldopen" id="foldopen00137" data-start="{" data-end="}">
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"><a class="line" href="group__shark__globals.html#ga964e703a6f0f014785ea68ce03cd7ad2">  137</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="group__shark__globals.html#gaf437ca980035fd6fb152bcfa0cd34d82" title="rotates a matrix using a householder reflection">applyHouseholderOnTheLeft</a>(</div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno">  138</span>    matrix_expression&lt;Mat, Device&gt;&amp;&amp; matrix,</div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno">  139</span>    vector_expression&lt;R, Device&gt; <span class="keyword">const</span>&amp; reflection, </div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno">  140</span>    T <span class="keyword">const</span>&amp; beta</div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno">  141</span>){</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno">  142</span>    <a class="code hl_function" href="group__shark__globals.html#gaf437ca980035fd6fb152bcfa0cd34d82" title="rotates a matrix using a householder reflection">applyHouseholderOnTheLeft</a>(matrix(),reflection,beta);</div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno">  143</span>}</div>
</div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno">  144</span><span class="comment"></span> </div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno">  145</span><span class="comment">/// \brief Initializes a matrix such that it forms a random rotation matrix.</span></div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno">  146</span><span class="comment">///</span></div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno">  147</span><span class="comment">/// The matrix needs to be quadratic and have the proper size</span></div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno">  148</span><span class="comment">/// (e.g. call matrix::resize before).</span></div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno">  149</span><span class="comment">///</span></div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno">  150</span><span class="comment">/// One common way to  do this is using Gram-Schmidt-Orthogonalisation </span></div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno">  151</span><span class="comment">/// on a matrix which is initialized with gaussian numbers. However, this is</span></div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno">  152</span><span class="comment">/// highly unstable for big matrices. </span></div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno">  153</span><span class="comment">///</span></div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno">  154</span><span class="comment">/// This algorithm is implemented from one of the algorithms presented in</span></div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno">  155</span><span class="comment">/// Francesco Mezzadri &quot;How to generate random matrices from the classical compact groups&quot;</span></div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno">  156</span><span class="comment">/// http://arxiv.org/abs/math-ph/0609050v2</span></div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno">  157</span><span class="comment">///</span></div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno">  158</span><span class="comment">/// He gives two algorithms: the first one uses QR decomposition on the random gaussian</span></div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno">  159</span><span class="comment">/// matrix and ensures that the signs of Q are correct by multiplying every column of Q</span></div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno">  160</span><span class="comment">/// with the sign of the diagonal of R. </span></div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno">  161</span><span class="comment">///</span></div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno">  162</span><span class="comment">/// We use another algorithm implemented in the paper which works similarly, but </span></div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno">  163</span><span class="comment">/// reversed. We apply Householder rotations H_N H_{N-1}..H_1 where</span></div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno">  164</span><span class="comment">/// H_1 is generated from a random vector on the n-dimensional unit sphere.</span></div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno">  165</span><span class="comment">/// this requires less operations and is thus preferable. Also only half the</span></div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno">  166</span><span class="comment">/// random numbers need to be generated</span></div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno">  167</span><span class="comment"></span><span class="keyword">template</span>&lt; <span class="keyword">class</span> MatrixT&gt;</div>
<div class="foldopen" id="foldopen00168" data-start="{" data-end="}">
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"><a class="line" href="group__shark__globals.html#gaca87daf2818038a5ad7f7b89f0d60d3b">  168</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="group__shark__globals.html#gaca87daf2818038a5ad7f7b89f0d60d3b" title="Initializes a matrix such that it forms a random rotation matrix.">randomRotationMatrix</a>(random::rng_type&amp; rng, matrix_container&lt;MatrixT, cpu_tag&gt;&amp; matrixC){</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno">  169</span>    MatrixT&amp; matrix = matrixC();</div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno">  170</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(matrix.size1() == matrix.size2());</div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno">  171</span>    <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(matrix.size1() &gt; 0);</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno">  172</span>    <span class="keywordtype">size_t</span> size = matrix.size1();</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno">  173</span>    diag(matrix) = repeat(1.0,size);</div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno">  174</span> </div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno">  175</span>    RealVector v(size);</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno">  176</span>    <span class="comment">//we skip the first dimension as the rotation of a 1d vector is just the identity</span></div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno">  177</span>    <span class="keywordflow">for</span>(std::size_t i = 2; i != size+1;++i){</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno">  178</span>        <span class="comment">//create the random vector on the unit-sphere for the i-dimensional subproblem</span></div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno">  179</span>        <span class="keywordflow">for</span>(std::size_t j=0;j != i; ++j){</div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno">  180</span>            v(j) = <a class="code hl_function" href="namespaceshark_1_1random.html#a972c5f7f031612a130aa077fc9136a9f" title="Draws a number from the normal distribution with given mean and variance by drawing random numbers fr...">random::gauss</a>(rng);</div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno">  181</span>        }</div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno">  182</span>        subrange(v,0,i) /=norm_2(subrange(v,0,i));<span class="comment">//project on sphere</span></div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno">  183</span>        </div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno">  184</span>        <span class="keywordtype">double</span> v0 = v(0);</div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno">  185</span>        v(0) += v0/std::abs(v0);</div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno">  186</span>        </div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno">  187</span>        <span class="comment">//compute new norm of v</span></div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno">  188</span>        <span class="comment">//~ double normvSqr = 1.0+(1+v0)*(1+v0)-v0*v0;</span></div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno">  189</span>        <span class="keywordtype">double</span> normvSqr = norm_sqr(subrange(v,0,i));</div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno">  190</span>        </div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno">  191</span>        <span class="comment">//apply the householder rotation to the i-th submatrix</span></div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno">  192</span>        <a class="code hl_function" href="group__shark__globals.html#gaf437ca980035fd6fb152bcfa0cd34d82" title="rotates a matrix using a householder reflection">applyHouseholderOnTheLeft</a>(subrange(matrix,size-i,size,size-i,size),subrange(v,0,i),2.0/normvSqr);</div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno">  193</span>        </div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno">  194</span>    }</div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno">  195</span>}</div>
</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno">  196</span><span class="comment"></span> </div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno">  197</span><span class="comment">//! Creates a random rotation matrix with a certain size using the random number generator rng.</span></div>
<div class="foldopen" id="foldopen00198" data-start="{" data-end="}">
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"><a class="line" href="group__shark__globals.html#gaf8329c98af0ef6bed47d41ea50552885">  198</a></span><span class="comment"></span>RealMatrix <a class="code hl_function" href="group__shark__globals.html#gaca87daf2818038a5ad7f7b89f0d60d3b" title="Initializes a matrix such that it forms a random rotation matrix.">randomRotationMatrix</a>(random::rng_type&amp; rng, <span class="keywordtype">size_t</span> size){</div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno">  199</span>    RealMatrix mat(size,size);</div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno">  200</span>    <a class="code hl_function" href="group__shark__globals.html#gaca87daf2818038a5ad7f7b89f0d60d3b" title="Initializes a matrix such that it forms a random rotation matrix.">randomRotationMatrix</a>(rng, mat);</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno">  201</span>    <span class="keywordflow">return</span> mat;</div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno">  202</span>}</div>
</div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno">  203</span> </div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno">  204</span><span class="comment"></span> </div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno">  205</span><span class="comment">/** @}*/</span></div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno">  206</span>}}</div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno">  207</span><span class="preprocessor">#endif</span></div>
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